Q. 440 - Intermediate Algebra

Question

Find: (a) the zeros of the function. (b) the $x$-intercepts of the graph of the function. (c) the $y$-intercept of the graph of the function.

$f\left(x\right)=6x^2-13x-5$

Step-by-Step Solution

Answer

(a) The zeroes of the function are $x=\frac{5}{2},-\frac{1}{3}$

(b) The $x$-intercepts of the graph of the function are  $\left(\frac{5}{2},0\right),\left(-\frac{1}{3},0\right)$

(c) The $y$-intercepts of the graph of the function are $(0,-5)$

Step 1. Given

The function is $f\left(x\right)=6x^2-13x-5$

Step 2. (a) Find the zeroes of the function.

For zeroes of the function, substitute $f\left(x\right)=0$.

 $$\begin{gathered} f\left(x\right)=6x^2-13x-5 \\ 0=6x^2-13x-5 \\ 6x^2-15x+2x-5=0 \\ 3x(2x-5)+1(2x-5)=0 \\ (2x-5)(3x+1)=0 \end{gathered}$$

So,

$x=\frac{5}{2},-\frac{1}{3}$

Step 3. (b) Find the x -intercepts of the graph of the function.

Put $f\left(x\right)=0$ to find $x$-intercept.

$f\left(x\right)$ is $0$ at $x=\frac{5}{2},-\frac{1}{3}$

The $x$-intercept of the graph of the function are $\left(\frac{5}{2},0\right),\left(-\frac{1}{3},0\right)$

Step 4. (c) Find the y -intercepts of the graph of the function.

Put $x=0$ to find the $y$-intercept.

$$\begin{gathered} f\left(x\right)=6x^2-13x-5 \\ f\left(0\right)=6{\left(0\right)}^2-13\left(0\right)-5 \\ f\left(0\right)=-5 \end{gathered}$$

The $y$-intercept of the graph of the function is $(0,-5)$